C12 technical supplement scientific notation 2 density, unit conversion 3 gravitational force 4 light:wavelengths, etc. 5 thermal radiation (overview) 6 thermal radiation (stefan-boltzmann law) 7 thermal radiation (wein s displacement law) 8 ellipses 9 spheres 9 kepler s laws: #1 10 kepler s laws: #2 11 kepler s laws: #3 12 atmospheric escape 13 vapor saturation 14 scale height 16 UC Berkeley 2007-07-22 mikewong
scientific notation Scientific notation is a convenient way to represent small and large numbers. The exponent (power of ten) can also be thought of as the number of digits to hop for the decimal point. A positive exponent means hop to make a bigger number, and a negative exponent means hop to make a smaller number. Take these two numbers for example: a = 9.3 10 7 b = 2.15 10 3 To convert a to a regular number, the exponent of 7 tells you to bounce the decimal point seven spaces to the right. One bounce (93.), two bounces (930.),... six bounces (9 300 000.), seven bounces (93 000 000.). So a is 93 million. For b, you would bounce three digits to the left, to make a smaller number: 0.00155. Math with scientific notation is easy. To calculate a b with the values given, you can first multiply 9.3 2.15 = 20, and then to find the exponents, just add 7 3 = +4. So the answer would be: a b = 2.0 10 5, or 20 thousand. For division, a b = 9.3 2.15, with an exponent of 7 3 = 7 + 3 = 10, so: a b = 4.3 10 10, or 43 billion. In this last example, my calculator actually told me that 9.3 2.15 = 4.32558, but I just wrote 4.3. Why? Because I m just lazy? No, because a only has two significant digits (the 9.3 before the exponent), so the answer can only have two significant digits. 2
density Density is mass divided by volume, so the formula is: ρ = m V where m is the mass, V is the volume, and density is represented by the Greek letter r (rho). Here are some densities of common solar system materials: water 1 g/cm 3 rock 2700 kg/m 3 iron 7.9 10 3 kg/m 3 When working with densities (or anything else) make sure you pay attention to the units of measurement! For example, at first it looks like rock is about 3000 times denser than water. To check this, we can convert the densities so that they are in the same units. Only then can we compare them. Converting units is easy if you remember to just multiply by one, because 1 x = x. One can be written many different ways. For example, 1 kg / 1000 g is one way of writing one, because 1000 g = 1 kg. Use this to convert the density of water from g/cm 3 into kg/m 3 : 1 g/cm 3 1 kg 1000 g = 0.001 kg/cm3 100 cm 100 cm 100 cm 1 m 3 = 1000 kg/m 3 3
gravitational force There is a gravitational force attracting any two massive objects. The force depends only on the masses and their separation from each other: F = Gm 1m 2 r 2, where F is the force, G is the gravitational constant, m 1 and m 2 are the masses of the two objects, and r is the distance between them. This expression is known as Newton s Law of Gravitation. If you use a value of G = 6.67 10 11 N m 2 / kg 2, then make sure you use matching units in the rest of the formula: r in meters, m in kg, and F in N (Newtons, a unit of force). 4
light: wavelengths, etc. Light can thought of as a wave or a particle (called a photon). We characterize light by its color. A particle is characterized by its energy. A wave is characterized by its wavelength, frequency, and speed. So wavelength or photon energy are ways of measuring the color of light. Light always travels at a constant speed in a vacuum. We call this speed c. A wave is a periodically repeating function. This diagram shows that the wavelength is the distance between two troughs (or two crests) of the wave: crest wavelength If you imagine a stationary observer being passed by this wave, which travels at the speed c, then the observer would measure the period of the wave to be the time it takes for one whole wave cycle (crest to crest) to pass by. If the observer measured how many cycles passed by within one second, she would know the frequency of the wave. The wavelength and the frequency of light are related by this formula: λ ν = c, trough where l (pronounced lambda) is the wavelength, and n (pronounced nu) is the frequency. Instead of describing the wavelength of light, you could describe the energy of its photons. For light of a certain wavelength l, its photons have energy E: E = h c/λ, where h is a constant called the Planck number. 5
thermal radiation (overview) Thermal radiation is given off by matter depending on its temperature and other properties. We often discuss ideal thermal radiators, or blackbodies, which absorb and emit radiation perfectly at all wavelengths. The spectrum given off by a blackbody depends only on its temperature and area. Thermal radiation has a continuous spectrum. Light is given off at all wavelengths across a wide range of the spectrum. Here are thermal emission spectra for objects at four different temperatures: 10 10 6000 K Intensity 10 6 10 2 2500 K 730 K 300 K 10 2 10 2 10 3 10 4 10 5 Wavelength (nm) The plot demonstrates two key properties of thermal radiation. Higher-temperature objects radiate more energy than lowertemperature objects. Also, the wavelength of maximum emission shifts to longer wavelengths as temperature decreases. What would these spectra look like to our eyes? The visible range falls between the dashed lines at 400 nm (blue) and 700 nm (red). The 6000-K object, close to the temperature of the Sun, would look basically white, since its spectrum is flat within the visible range. The 2500-K object would look much redder, because the intensity of its red radiation is higher than the intensity of its blue radiation. The thermal radiation from the colder objects would not be visible to us, since they give off most of their energy at much longer infrared wavelengths. 6
thermal radiation (stefan-boltzmann) The Stefan-Boltzmann law is a simple equation which says that hotter (higher temperature) objects give off more thermal radiation: E = s T 4, where the thermal energy per unit area per unit time is E, the temperature in Kelvins is T, and s (pronounced sigma) is the Stefan-Boltzmann constant, 5.67 10 8 J s 1 m 2 K 4. The graph of thermal spectra shows the intensity at any wavelength. E from the Stefan-Boltzmann law just comes from adding up the intensity at all wavelengths, so you can think of E as the area under the curve. The area under the 6000-K curve is obviously greater than the area under the 300-K curve, and the difference is actually quite large because the y-axis is on a logarithmic scale. 10 10 6000 K Intensity 10 6 10 2 2500 K 730 K 300 K 10 2 10 2 10 3 10 4 10 5 Wavelength (nm) 7
thermal radiation (wein s displacement law) For a thermally-emitting body, there s always a certain wavelength where the intensity of the thermal radiation is greatest. This wavelength, l max, is given by Wein s Displacement Law: λ max T = 2900 µm K = 0.29 cm K = 2.9 10 7 Å K The constant on the right takes different values, depending on the units of wavelength you re using. Using Wein s law, a spectrum recorded from a thermal radiator becomes a thermometer capable of measuring the object s temperature! Just determine l max from the spectrum, and use Wein s law to get the temperature. 8
ellipses An ellipse is a geometrical figure resembling a flattened circle. The amount of flattening is given by the eccentricity, e, of the ellipse. A circle is a special case of an ellipse with an eccentricity of 0. Flatter ellipses have greater eccentricities, with 1 being the maximum. Here are the parts of an ellipse: a b The two dots are the foci (singular: focus) of the ellipse. This ellipse has a pretty high eccentricity of 0.87, so the foci are far from the center. For less eccentric ellipses, the foci would be closer to the center, and for a circle, the foci are actually at the center. The large axis, a, is called the semi-major axis, and b is the semi-minor axis. For a circle, a = b. spheres Spheres are simple respresentations of the shapes of planets, plutoids, stars, raindrops, sand grains, chondrules, tektites, and many other things. It s useful to be able to calculate the surface area and volume of spheres. The area is: A = 4π r 2, and the volume is: V = 4π r3 3. 9
kepler s laws: #1 Planetary orbits are in the shape of an ellipse, with the sun at one focus. Since the sun is at a focus (not at the center), the planet s distance from the sun will change with time. The point of closest approach to the sun is called perihelion, and the farthest point along the orbit from the sun is aphelion: aphelion / apoapsis Sun perihelion / periapsis Kepler s laws apply to the planets (and comets, asteroids, Plutinos, plutoids, etc.) revolving around the sun, but they also describe many other situations, including moons orbiting planets, exoplanets orbiting their stars, and even stars orbiting the supermassive black hole at the center of the Milky Way Galaxy. Perihelion and aphelion only refer to orbits around the sun. Orbits around the Earth have a perigee and an apogee. Periapsis and apoapsis are the general terms that are correct for any orbit. 10
kepler s laws: #2 Over a given period of time, a straight line connecting a planet and the sun will sweep out a constant area. 3 2 1 4 Let s say a comet goes from point 1 to point 2 in two months, and from point 3 to point 4 in two months. Then Kepler s second law says the two grey areas are equal. For this to be true, the distance from 1 to 2 must be less than the distance from 3 to 4. Since velocity = distance / time, and we re using the same time interval in both cases, this also means that the velocity of the comet at 1 is less than the velocity at 3. 11
kepler s laws: #3 The square of a planet s orbital period is proportional to the cube of its orbital semimajor axis. We can write this mathematically either with the proportionality symbol ( ), or with a constant of proportionality k, or in a general form that is true for any orbital system: P 2 a 3 P 2 = k a 3 P 2 = 4π 2 a 3 G M total In the first form, you can use whatever units you want. In the second form, the units are set by the value of k. For objects in orbit around the sun, k = 1 year 2 / AU 3 works great. An AU, or astronomical unit, is the distance from the Earth to the sun, or 1.5 10 11 m. The third form is derived from Newton s Law of Gravitation, and is true for any orbital system. In this form, the units must match those of G, which is 6.67 10 11 N m 2 / kg 2. Since 1 N = 1 kg m /sec 2, P will be in units of seconds. 12
atmospheric escape Why don t atmospheres escape? They do, sometimes. Maintaining at atmosphere is a balance of forces. Gravity acts to pull atmospheric gases to the planet, but the little gas molecules are bouncing around due to their thermal energy, and some of them may bounce fast enough to escape the planet. This would be thermal escape, usually the strongest escape mechanism (although there are others). A rough rule of thumb is that an atmosphere will experience thermal escape if the average thermal velocity of the gas molecules is greater than about one sixth of the escape velocity of the planet: 6 v = v esc., where the average thermal velocity of the gas molecules is 8RT v = πm = 8RT, πµm u and T is the temperature in K, R is the molar gas constant (8.31 J K 1 mol 1 ), M is the molar mass of the gas (in kg mol 1 ), µ is the atomic or molecular weight of the gas (just the number of protons and neutrons in each particle), and M u is the molar mass constant of 0.001 kg mol 1. M and µm u are two different ways of representing the mass of the gas particles. The escape velocity from a planet or other object is v esc. = 2 g r = 2Gmplanet where G is the gravitational constant, m is the mass of the planet, g is the gravitational acceleration of the planet, and r is the distance to the planet. r, 13
vapor saturation Depending on conditions such as temperature and pressure, matter can exist in one or more phases: gas, liquid, or solid. At a given temperature, some amount of gas can coexist in equilibrium with a solid or liquid phase. To illustrate how this works, we ll consider the familiar example of a plastic container and some rice. This requires understanding partial pressure. If you have a gas of mixed composition, such as moist air from Earth, partial pressure just describes the fraction of the total pressure that is contributed by each component of the gas. Moist air is about 78% nitrogen (N 2 ), 21% oxygen (O 2 ), and less than 1% water vapor (H 2 O). Then if the total pressure is about typical sea level pressure (1 bar = 10 5 Pa), then the partial pressure of oxygen is just 0.21 10000 = 2100 Pa. Imagine putting some hot rice in a plastic container, while in a room with completely dry air with 0% relative humidity (like desert air). The rice will gradually dry out, as hot liquid water within the rice evaporates (changes into vapor or gas). If you put a lid on the container, the rice will continue to dry out, but it will stop drying out when enough water evaporates so that the partial pressure of the water vapor equals the saturation vapor pressure of water at that temperature. At that point, we say that the air in the box is saturated, or that the relative humidity is 100%. The curve below shows how the saturation vapor pressure of water changes as a function of temperature: 14
vapor saturation (continued) Now consider the tightly-closed plastic container with rice inside. We left it closed long enough for the water partial pressure to increase up to the saturation vapor pressure. If we then stick the box in the refrigerator, we ll come back later to find that the lid of the box is covered with little water droplets. Why? When we chilled the container, the temperature of the gas inside dropped. The curve on the previous page showed that as temperature decreases, the saturation vapor pressure decreases. That means the partial pressure of water vapor in the box becomes higher than the saturated vapor pressure, a non-equilibrium condition. To restore equilibrium, water starts condensing out from vapor to liquid, reducing the partial pressure of water vapor until it s back down to the saturated vapor pressure. That s also why the lid may be stuck on tightly after the water condenses out. Condensation drops the partial pressure of water, but it also drops the total pressure in the box by the same amount, forming a weak vacuum seal. Relative humidity was mentioned on the previous page. Relative humidity is the vapor partial pressure divided by the saturation vapor pressure. This reaches a maximum of 1 (or 100%) when saturation is reached, and a minimum of 0 when there is no vapor present. Relative humidities can sometimes exceed 100%, in a condition called supersaturation, but this non-quilibrium state is temporary, and it ends when enough vapor condenses to bring the partial pressure back down to the saturation vapor pressure. 15
scale height The scale height, H, describes how quickly atmospheric density and pressure decrease with height in an object s atmosphere. If we ignore some complexities like temperature variation, then the variation of pressure as a function of height is p = p 0 exp ( z/h), where p 0 is the pressure at a reference level (like at the surface), and z is the height above that level. At the Earth s surface, the scale height is around 7 km. So the pressure at a height of 7 km would be about p 0 exp( 7 km / 7 km) = 1 bar e 1 = 0.37 bar. The scale height can be calculated as H = k T m g = k T µ m u g, with k the Boltzmann constant (1.38 10 23 J / K), T the temperature (in Kelvins of course), m the average mass of atmospheric molecules (in kg), g the gravitational acceleration (in m s 2 ), µ the average molecular weight of the atmosphere (just a number with no units), and m u the atomic mass constant (1.66 10 27 kg). The scale height is a measure of how compact or extended an atmosphere is, so this expression shows that higher temperatures or lower gravity results in more extended atmospheres, while heavier gases mean more compact atmospheres. 16